Aharanov-Bohm electron phase shift

[Norman Albers]

The Aharanov-Bohm effect predicted the phase-shift of electron interference patterns from a split beam passing either side of a magnetized iron whisker. It is supposed that there are no magnetic fields present outside the solenoid, since we can make them arbitrarily small with a "longer" solenoid or whisker. I claim we can Lorentz-transform the problem and show that in the electrons' frames of reference, what was a charge-neutral stack of circular current (solenoid), now has an electric dipole field which clearly acts on the passing electrons! Consider a slice of the solenoid looking down at the circle. We assign current circulating CCW, and consider electrons passing below and above the circle. The B-field implied inside the coils comes upward "out of the paper". The electrons move at relativistically 'low' velocity from left to right. At the lower part of the loop, there is positive [math]j_x[/math] while current at the top points in <-x>. When we transform the source 4-vector, we develope charge densities of [math]-\beta\gamma j_x[/math], while the [math]j_x[/math] itself gets multiplied by [math]\gamma[/math]. The point here is that the charge density is negative on the bottom and positive on top. Thus there is developed a net dipole field which pulls down electrons above the circle, and also pushes down those below. If my calcs are correct, then what was usually expressed as a quantum phase change of [math]q/\hbar \int A\cdot ds[/math] can also be figured from the expression of the electric dipole field, since scalar potential gives phase change of [math] -q/\hbar \int \Phi dt [/math]. The vector potential term is the total flux contained in the stack (and in each loop cross-section not near the ends). The scalar integration can be done for one loop and then integrated up and down the stack for the total contribution. These are coming out the same within a factor of [math]\pi[/math] or something like that, having to do with the characterization of the dipole moment. This will remain with an arbitrarily small diameter for the same reason that we demand a certain magnetic flux even as we shrink the whisker. Both are proportional to current and diameter.

[Norman Albers]

The farfield of a static electric dipole goes as [math] r^{-3 }cos\theta[/math]. Thus its potential goes as [math]r^{-2}cos\theta [/math]. When we integrate this source field "up and down the solenoid stack", another order of r is reduced and we have an [math] r^{-1}[/math] dependence just as we do from a "solenoid" vector potential... Once you know you have this form, the difference of the integrals of paths above and below are the loop integral and are dependent only on the internal source. B-B-BINGO, you can look at it as mystic QM, or you can see it as electric force transmitted through a purely electric virtual vacuum field.

[Farsight]

That sounds darn interesting, Albers. I drew myself a picture and found one on the internet.

 

http://fisica.udea.edu.co/~mpaez/aharanov/Bohmen.html

 

I'm not clear on your second post. What's the BINGO for? I'm scratching my head a little because I've never even heard about about this Aharanov-Bohm effect. It reminds me a little o the Hall Effect, but nevermind. Have you nailed some bit of mystic QM crap? Funnily enough I was on "another forum" looking at a quantum entanglement thread thinking I'm going to write an essay called QUANTUM ENTANGLEMENT EXPLAINED. I've been writing SPACE EXPLAINED, it needs a polish. But you know your purely electric virtual vacuum field? Maybe it's a little more common than you think.

 

Oooh. Wait a minute Albers. Where is this force transmitted from? From the coil to the electrons? Or have you just rumbled AntiGravity?

[Norman Albers]

The point is that in the electron's frame of reference there is an electric dipole field from the solenoid. I wouldn't go so far as to critically say "crap" because Quantum Mechanics successfully psyches out so much phenomonology. Furthermore I have learned to work primarily with the 4-vector potential as opposed to E- and B-fields. This fascinates me because both perspectives seem to be needed. Any loop current (including that of a particle magnetic moment) manifests a dipole electric field under Lorentz velocity transform. Farsight, have you read of the vacuum being considered as purely electric? Talking this past winter I challenged Puthoff who still figured virtual manifestations had to have spin to transmit magnetic forces. I think not but am still working to picture or mathematically command the Lorentz transform of the virtual medium. . . . . . . . . . . What I have presented hangs together nicely in terms of path integrals and "phase" but I do not yet understand implications of momentum. This field picture yields an impulse downwards and yet the interference pattern shifts upwards. We are still doing wave mechanics here, with double slits for starters, then with wave function phase. I am creatively confused.

[Farsight]

Noted Albers. Yep, just electric. And even that isn't quite the right word. Remember we were talking about what the right word was? Have a look at my recent posts on PhysOrgForum and see what I was saying to Darren. Puthoff is wrong. Aw dammit, I'm going to have to send you Charge Explained. It will give you the vision. Do you promise to a) keep it to yourself b) give me credit if it's due and c) not say a dicky bird about how refraction and superconduction works?

[Norman Albers]

The conclusion of the Aharanov-Bohm paper gives tantalizing teases. They informed me that the vector potential term in the momentum operator comes out working like an index of refraction(!!!) and use the word superconducting in their last line. In my photon paper I discovered that if there are wave packets with Gaussian localization, this implies a superconducting response of reaction currents, identified as the first term in: [math]j=(-\lambda^2 +\rho/U)A[/math]. . . . . . As far as the dipole field I just cooked up, it seems spooky because there were neither electric or magnetic fields of significance in the lab frame, outside of the solenoid. The usual sense of Lorentz-transform of fields sources E's and B's into each other locally. Hmmm.

[Norman Albers]

Farsight, I said BINGO because the vector potential outside of a long solenoid is 1/r. Such a form of far-field is "path-independent", like if you consider larger and larger circles. In the electron's reference frame, I start with an electric dipole moment in each slice of the solenoid, integrated through the stack. This similarly comes out to 1/r in the far and so the difference between electrons, going by on top and below, is described by a loop integral which is determined only by the sources inside the loop, which is the solenoid as described. Thus one can figure the QM phase as either the electron moving through an A-field, or in the frame where it is at rest, the solenoid and its fields coming past. In the latter, the term [math]A\cdot ds[/math] is zero since ds describes path movement of the electron. We come out with the same phase shift generated by the scalar integral. . . . . .I still am mystified about the L-transform. I can say that I understand electric fields coming from [math]-\dot A[/math] .

[Farsight]

I've sent you CHARGE EXPLAINED Albers. It mentions refraction, and superconduction, and should give you "grasp" of what an electric field actually is. The relativity aspect is crucial.

[Norman Albers]

Thank you Farsight, I look forward to reading your ideas. I have reached a very cool conclusion about the solenoid fields experienced in the electron's frame of reference, passing by. We know that no fields are expressed, basically, in the lab frame. The electron does "see" a charge-field such as I described, on the near and far sides of the whisker. I calculated also the electric field due to the passing and changing A-field. Picture the circles of constant A, centered around the solenoid, and passing by your reference point. Once I got the vectors and integrations straight, this electric field: [math]E_A=-\dot A [/math] works out to be equal and opposite to that figured from the electric dipole potential. There is a factor of two but I think that's from the dipole strength involving only the x-components of the current in the loops, while magnetic flux feels both x,y-components. Thus the net electric field is still zero! I find this fascinating.

[Farsight]

I think I need a picture, Norm. But until then it sounds like what you've got is a spiral drill bit.

[Norman Albers]

You have to sit and do the Lorentz-transform of the current loop. Then you gotta hang out with the circular vector potential of the "long solenoid" being derived as 1/r. Last you must know from electrodynamics that generally electric fields: [math] E=-\nabla U -\dot A [/math] . What I am doing is examining and exhausting the electrodynamic essences in an essentially DeBroglie wave interference of two "closely placed slits". My friend solidspin agreed that there is a "change in quantum phase" equally to electron fields passing on top or, oppositely, on bottom of a magnetized whisker in between and after the slits, but he thought this was not directly tied to momentum change. I am not convinced of this yet, but we can agree there are no net electrodynamic forces. What I might offer as a semiclassical bailout, or picture, is that the lower electron is being slightly repelled, coming and going, so that its momentum is lower at close approach. This is associated with a longer wavelength DeBroglie wave, and the opposite is true of the deflection on the upper path. If I am correct, this corresponds to the phase shifts called for from quantum potential theory. If you looked only at the dipole electric interaction you'd think the electrons deflect downwards but they don't! We must admit to semiclassical language because we really consider the field of one electron passing through the system. I can say the semiclassical picture offers no field interactions to bend paths up or down, but the waves propagated through the two slits show a shift of interference probability upward, on a vertical screen of detectors on the right.

[Farsight]

I think the common interpretation of QM is set to change, Norm. Radically.

[Norman Albers]

I am glad to hear from Farsight that many other people are working in the belief that we can know more. Paradigms of understanding shift awkwardly. Back to my solenoid fields, in the electron's frame of reference, the electric field from the "induced charge field" on the solenoid would be slightly repulsive to the electron passing below the loops. There is, however, an equal and opposite electric field expressed by [math]-\dot A [/math] , as the circles of vector potential from the solenoid cruise by. This would, by itself, be taken as accelerative, attractive. Now to mess with our minds, we consider the momentum implied by any charge in an A-field, namely [math]\rho A[/math]. This points backward, repulsively. You just have to remember the basis of your accounting and not lose your nerve. . . . . . . My fundamental challenge is to render this elusive rotation of the vacuum field in terms of directional polarizability fields. I am workng to solve my Reissner-Nordstrom off-diagonal magnetic terms, actually a parallel problem...

[Norman Albers]

The final point about the Lorentz transform to the reference frame of the charge passing by, is that we derive equal and opposite "electric fields" from the scalar potential interaction, here a dipole electric field; and the time-changing A-field of the passing solenoid or magnetic whisker. Otherwise we are free to find truth.

[Farsight]

Read this one next, Norm:

 

REFERENCE FRAMES

[Norman Albers]

I have discussed how gravitation can be expressed as a thickening of the vacuum polarizability, which by itself is not a polarized state. On the other hand, charge and E&M fields are polar expressions, polarized dispositions of this same medium. My progress has been slow because I am being thorough: I have been digesting the mathematics of the Kerr rotating GR solution, and of the Lense-Thirring "low-field" expansion, as I hope now to construct a similar representation given a source of axially symmetric currents as per my inhomogeneous electron model. An adequate tensor expression should show how the vacuum field shows shear without any polarization, and without intrinsic spin, around a solenoid.